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$M$.

${\color{Green}X}=\{x_1,\ldots

y_j)$

Cartesian

edge.(By

Y\to\mathbb{R}$

y_j)$.

x_j)$.

\color{RubineRed}\text{pink}}$.

Y}$

right?There

Y\times Z$

Hilbert

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Nice

\color{RubineRed}Y}\to\mathbb{R}$

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Positivity 49.00%

Negativity 51.00%

SOURCE:
https://www.math3ma.com/blog/matrices-probability-graphs
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Then a matrix $M$ corresponds to a weighted bipartite graph in the following way: the vertices of the graph have two different colors provided by ${\color{Green}X}$ and ${\color{RubineRed} Y}$, and there is an edge between every $x_i$ and $y_j$ which is labeled by the number $M_{ij}$. For instance...Given two matrices (graphs) $M\colon {\color{Green}X}\times {\color{RubineRed}Y}\to\mathbb{R}$ and $N\colon {\color{RubineRed}Y}\times {\color{ProcessBlue}Z}\to\mathbb{R}$, we can multiply them by sticking their graphs together and traveling along paths: the $ij$th entry of $MN$, i.e. the value of the edge connecting $x_i$ to $z_j$, is obtained by multiplying the edges along each path from $x_i$ to $z_j$ and adding them together. It does so by way of another nice little fact: âFor example:The graphs of such probability distributions allow us to read off some nice things.By construction, the edges of the graph capture the joint probabilities: the probability of $(x_i,y_j)$ is the label on the edge connecting them.Marginal probabilities are obtained by summing along the rows/columns of the matrix (equivalently, of the chart above). Likewise, the conditional probability of some $y_i$ given $x_j$ is the value of the edge joining the two, divided by the sum of all edges incident to $x_j$.Pretty simple, right?There's nothing sophisticated here, but sometimes it's nice to see old ideas in new ways!I'll close this post with an addendumâanother simple fact that I think is really delightful, namely: matrix arithmetic makes sense over commutative rings. Because a matrix $M\colon X\times Y\to\mathbb{Z}_2$ is the same thing as a relation. In other words, every $\mathbb{Z}_2$-valued matrix defines a relation, and every relation defines a $\mathbb{Z}_2$-valued matrix: $M_{ij}= 1$ if and only if the pair $(x_i,y_j)$ is an element of the subset $R$, and $M_{ij}= 0$ otherwise.The graphs of matrices valued in $\mathbb{Z}_2$ are exactly like the graphs discussed above, except now the weight of any edge is either 0 or 1. If the weight is 0 then, as before, we simply won't draw that edge.(By the way, you can now ask, "Since every relation corresponds to a matrix in $\mathbb{Z}_2$, what do matrices corresponding to equivalence relations look like?" But I digress....)By changing the underlying (semi)ring from $\mathbb{R}$ to $\mathbb{Z}_2$, we've changed the way we interpret the weights.

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